Question

Nitrogen gas is expanded in a polytropic process with \(n=1.25\) from \(2 \mathrm{MPa}\) and \(1200 \mathrm{K}\) to \(200 \mathrm{kPa}\) in a piston-cylinder device. How much work is produced and heat is transferred during this expansion process, in \(\mathrm{kJ} / \mathrm{kg} ?\)

Short answer

Question: Determine the work done (W) and heat transfer (Q) during a polytropic process with a polytropic index of 1.25 for nitrogen gas, having an initial pressure of 2 MPa, an initial temperature of 1200 K, and a final pressure of 200 kPa. Express your answer in kJ/kg. Solution: First, apply the ideal gas equation and polytropic equation of state to find the initial and final specific volumes (v1 and v2) and the final temperature (T2). Then, calculate the work done (W) using the formula for a polytropic process. Finally, apply the first law of thermodynamics to find the heat transfer (Q). Make sure to convert your final answers to kJ/kg.

Step-by-step solution

Calculate Initial and Final Specific Volumes

To determine the specific volumes, use the ideal gas equation: \( Pv = RT \) Rearrange the equation to calculate the specific volume (\(v\)) and assume that the gas constant for nitrogen (\(R\)) can be found in a textbook or resource: \(v = \frac{RT}{P}\) First, we calculate the initial specific volume \(v_1\): \(v_1 = \frac{R T_1}{P_1}\) Similarly, for the final specific volume \(v_2\): \(v_2 = \frac{R T_2}{P_2}\) Note that we don't know the final temperature \(T_2\) yet. However, using the polytropic equation of state for both initial and final states: \(\frac{P_1 v_1^n}{P_2 v_2^n} = 1\) From here, we can obtain the relationship between \(v_1\) and \(v_2\): \(v_2 = v_1 \left( \frac{P_1}{P_2} \right)^{1/n}\) After finding \(v_2\), we can determine the final temperature \(T_2\) using the ideal gas equation: \( T_2 = \frac{P_2 v_2}{R}\)

Calculate Work Done

To calculate the work done during the polytropic process, we use the work formula for a polytropic process: \(W = \frac{m R (T_1 - T_2)}{n - 1}\) Using the values for \(T_1\), \(T_2\), and \(n\), we can calculate \(W\). Be sure to convert the result to kJ/kg.

Apply the First Law of Thermodynamics to Find Heat Transfer

The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system: \(\Delta U = Q - W\) For an ideal gas, we can express the change in internal energy as: \(\Delta U = m C_v (T_2 - T_1)\) Where \(C_v\) is the specific heat at constant volume for nitrogen, which can be found in a textbook or resource. Substituting this expression into the first law of thermodynamics, we have: \(m C_v (T_2 - T_1) = Q - W\) Now, we can solve for the heat transfer \(Q\): \(Q = m C_v (T_2 - T_1) + W\) By plugging in the values for \(m\), \(C_v\), \(T_1\), \(T_2\), and \(W\), we can calculate \(Q\). Finally, make sure to convert the result to kJ/kg.

Key concepts

Thermodynamics

Thermodynamics is the fundamental science that deals with the relationships between heat and other forms of energy. In the context of the exercise provided, we're looking at a polytropic process—a specific type of thermodynamic process that involves both work and heat transfer.

During a polytropic process, energy is conserved according to the first law of thermodynamics, which states that the total increase in the energy of a system is equal to the amount of energy added as heat minus the work done by the system. This law is key to understanding how energy is transformed and transferred during the expansion of nitrogen gas in the problem we're examining. By applying these principles, you can calculate crucial variables such as the work output and heat transferred during the process.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics given by the relationship Pv = RT, where P is the pressure, v is the specific volume, R is the specific gas constant, and T is the absolute temperature.

The Role of the Ideal Gas Law in the Polytropic Process

In a polytropic process involving an ideal gas, the ideal gas law provides a way to determine initial and final states of the gas. It allows us to calculate the specific volume of the nitrogen gas at the start and end of the process, and, with the polytropic relationship, find out how the volume changes over the course of the expansion. Understanding these concepts is critical to solve for the work done and the heat transfer during the process.

When the nitrogen gas expands, its pressure and temperature change. The ideal gas law helps to establish the correlation between these changes, allowing us to find the final temperature after knowing the final specific volume. This information is vital to determine the thermodynamic properties of the system and complete the solution for the problem at hand.

Work and Heat Transfer

Work and heat transfer are two fundamental concepts in thermodynamics closely related to energy changes in a system undergoing a process.

Calculating Work in a Polytropic Process

For the nitrogen gas expansion in the polytropic process, the work done (denoted as W) can be calculated using the formula W = mR(T_1 - T_2)/(n - 1). It's essential to recognize how the initial and final temperatures, gas constant, mass of the gas, and the polytropic index n influence the amount of work produced. In the given exercise, the work is the energy exerted by the expanding gas on the surroundings.

Understanding Heat Transfer

As for heat transfer, this refers to the energy that is transferred into or out of the system due to a temperature difference. The polytropic expansion process described involves a transfer of heat that can be calculated by serving the first law of thermodynamics. It typically involves understanding the change in internal energy (ΔU) of the gas and the work done. By calculating work and applying the first law, it is possible to determine the heat transfer Q, which is critical for engineering applications where energy efficiency and thermal management are essential.